Think
back to when you were first introduced to functions, thin lines depicting a
single value output for each input in a domain.

A Function |

Compacting more data into inputs and
outputs provides not only more information, but also a more stunning
visualization of data. In a vector field, a single point can contain
information about location, strength and direction of a force. The more
information a function tracks, the more stunning the display becomes, with 4 or
even 5 dimensions represented on a graph of three-dimensional space and color.

Vectors depicting the strength and direction of a magnetic field at discrete points. |

A 3D graph, with a 4th color dimension |

With developments in technologies that offer efficient data
manipulation, the possibilities of what we can do with functions and data are
more far-reaching than ever. Anne M. Burns of Long Island University Uses
computers to create beautiful representations of functions.

Burns plots complex valued functions as a vector field, seen here. |

Attributing more dimensions to an occurrence is useful and
can be beautiful, but what if the object or function in question is impossible
to make sense of as it is? It is often handy to project or unfold an N-dimensional
surface onto an (N-1)-dimensional surface. Most of the time, in calculus, a
three-dimensional surface will be looked at as a two-dimensional projection on
the xy, yz, or xz plane in order to set up an integral to find the volume of
the object. In theoretical physics, this technique of reducing the dimension of
mysterious happenings is used to speculate the nature of the universe. A common
example, and perhaps the most accessible way to think of this process is the
unfolding of a four-dimensional cube, the tesseract.

The nets of a 3D cube and a 4D hypercube above. |

Dali's Corpus Hypercubus (1954) |

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At its core, mathematics does not only seek knowledge, but
also pursues beauty in the natural world.

__Works Cited__

- The Function graphic was found on the page of Maret School's BC calculus page, and is spliced with Charlie Brown of Peanuts, created by Charles M. Schulz.
- The Magnetic field graphic was found on Vassar College's Wordpress blog, under a lecture by Prof. Magnes.
- 4D graph curtesy of user Blue7 on math.stackexchange.
- Find all of Anne M. Burn's Work here.
- The Cube net image was found here.

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